Asymptotic Behaviour of an Empirical Nearest‐Neighbour Distance Function for Stationary Poisson Cluster Processes

Summary. For stationary POISSON cluster processes (PCP's) O on R the limit behaviour, as v(D) → ∞, of the quantity , where χ(x, r) = 1, if O(b(x, r)) = 1, and χ(x, r) = 0 otherwise, is studied. A central limit theorem for fixed r > 0 and the weak convergence of the normalized and centred empirical process on [0, R] to a continuous GAUSSian process are proved. Lower and upper bounds for the nearest neighbour distance function P1({φ:Y(b(0,r))≧1}) of a stationary PCP are given. Moreover, a representation of higher order Palm distributions of PCP's and a central limit theorem for m-dependent random fields with unbounded m are obtained. Both these auxiliary results seems to be of own interest.

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